Diffusion term
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Discretisation of the Diffusion Term
Description
For a general control volume (orthogonal, non-orthogonal), the discretization of the diffusion term can be written in the following form
where
- S denotes the surface area of the control volume
denotes the area of a face for the control volume
As usual, the subscript f refers to a given face. The figure below describes the terminology used in the framework of a general non-orthogonal control volume
A general non-orthogonal control volume
Note: The approaches those are discussed here are applicable to non-orthoganal meshes as well as orthogonal meshes.
A control volume in mesh is made up of set of faces enclosing it. Where represents the magnitude of area of the face. And n represents the normal unit vector of the face under consideration.
If 
and 
are position vector of centroids of cells P and N respectively. Then, we define
We wish to approaximate the diffusive flux 
at the face.
Approach 1
A first approach is to use a simple expression for estimating the gradient of a scalar normal to the face.
![D_f = \Gamma _f \nabla \phi _f \cdot \vec S_f = \Gamma _f \left[ {\left( {\phi _N - \phi _P } \right)\left| {{{\vec S_f} \over {\overrightarrow{d_{PN}}}}} \right|} \right]](/W/images/math/2/2/f/22f7fe72fd16546ff2d65bf48adf7a35.png)
where 
is a suitable face average.
This approach is not very good when the non-orthogonality of the faces increases. If this is the case, it is advisable to use one of the following approaches.
Approach 2
We define the vector
giving us the expression:
![D_f = \Gamma _f \nabla \phi _f \cdot{\rm{\vec S_f = }}\Gamma _{\rm{f}} \left[ {\left( {\phi _N - \phi _P } \right)\vec \alpha \cdot {\rm{\vec S_f + }}\bar \nabla \phi_f \cdot {\rm{\vec S_f - }}\left( {\bar \nabla \phi_f \cdot {\overrightarrow{d_{PN}}}} \right)\vec \alpha \cdot {\rm{\vec S_f}}} \right]](/W/images/math/7/c/f/7cf1e0cce420c5d8f260ea5c7f3340ff.png)
where 
and are suitable face averages.
Orthogonal Correction Approaches
In non-orthogonal grids, the gradient direction that will yield an expression involving the values at the neighboring control volumes will have to be along the line joining the centroids of the two control volumes. If this direction has a unit vector denoted by 
then, by definition
then the gradient in the direction of can be written as
If the surface vector 
is written as the summation of two vectors 
and 
where is in the direction joining the centroids of the two control volumes, we will then be able to express the diffusive flux in terms of the neighboring control volumes plus an additional correction. This is done as follows
.... (where E is the magnitude of 
At the outset, one obtains
The first term in the above equation can be thought of as the orthogonal contribution to the diffusive flux, while the second term represents the non-orthogonal effects. At this point, the vector has not been defined yet. There are three main methods to define this vector.
Minimum Correction
In the minimum correction approach, the vectors are defined as
Minimum Correction Approach
Orthogonal Correction
In the orthogonal correction approach, the vectors are defined as
Orthogonal Correction Approach
Over Relaxed Correction
Finally, in the over relaxed approach, we define
Over Relaxed Correction Approach
References
- Ferziger, J.H. and Peric, M. (2001), Computational Methods for Fluid Dynamics, ISBN 3540420746, 3rd Rev. Ed., Springer-Verlag, Berlin..
- Hrvoje, Jasak (1996), "Error Analysis and Estimation for the Finite Volume Method with Applications to Fluid Flows", PhD Thesis, Imperial College, University of London (download).
- Saad, Tony (2005), "Implementation of a Finite Volume Unstructured CFD Solver Using Cluster Based Parallel Computing", Thesis, American University of Beirut.
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